Suppose that Z is a random closed subset of the hyperbolic plane H-2, whose law is invariant under isometrics of H-2. We prove that if the probability that Z contains a fixed ball of radius 1 is larger than some universal constant p0 < 1, then there is positive probability that Z contains (hi-infinite) lines. We then consider a family of random sets in H-2 that satisfy some additional natural assumptions. An example of such a set is the covered region in the Poisson Boolean model. Let f(r) be the probability that a line segment of length r is contained in such a set Z. We show that if f(r) decays fast enough, then there are as. no lines i Z. We also show that if the decay of f (r) is not too fast, then there are as. lines in Z. In the case of the Poisson Boolean model with balls of fixed radius R we characterize the critical intensity for the as. existence of lines in the covered region by an integral equation. We also determine when there are lines in the complement of a Poisson process on the space of lines in H-2.

Dela på webben

Skapa referens, olika format (klipp och klistra)

BibTeX @article{Benjamini2009,author={Benjamini, I. and Jonasson, Johan and Schramm, O. and Tykesson, Johan},title={Visibility to infinity in the hyperbolic plane, despite obstacles},journal={Latin American Journal of Probability and Mathematical Statistics},issn={1980-0436},volume={6},pages={323-342},abstract={Suppose that Z is a random closed subset of the hyperbolic plane H-2, whose law is invariant under isometrics of H-2. We prove that if the probability that Z contains a fixed ball of radius 1 is larger than some universal constant p0 < 1, then there is positive probability that Z contains (hi-infinite) lines. We then consider a family of random sets in H-2 that satisfy some additional natural assumptions. An example of such a set is the covered region in the Poisson Boolean model. Let f(r) be the probability that a line segment of length r is contained in such a set Z. We show that if f(r) decays fast enough, then there are as. no lines i Z. We also show that if the decay of f (r) is not too fast, then there are as. lines in Z. In the case of the Poisson Boolean model with balls of fixed radius R we characterize the critical intensity for the as. existence of lines in the covered region by an integral equation. We also determine when there are lines in the complement of a Poisson process on the space of lines in H-2.},year={2009},keywords={continuum percolation, phase transitions, hyperbolic geometry, exceptional lines },}

RefWorks RT Journal ArticleSR PrintID 172144A1 Benjamini, I.A1 Jonasson, JohanA1 Schramm, O.A1 Tykesson, JohanT1 Visibility to infinity in the hyperbolic plane, despite obstaclesYR 2009JF Latin American Journal of Probability and Mathematical StatisticsSN 1980-0436VO 6SP 323OP 342AB Suppose that Z is a random closed subset of the hyperbolic plane H-2, whose law is invariant under isometrics of H-2. We prove that if the probability that Z contains a fixed ball of radius 1 is larger than some universal constant p0 < 1, then there is positive probability that Z contains (hi-infinite) lines. We then consider a family of random sets in H-2 that satisfy some additional natural assumptions. An example of such a set is the covered region in the Poisson Boolean model. Let f(r) be the probability that a line segment of length r is contained in such a set Z. We show that if f(r) decays fast enough, then there are as. no lines i Z. We also show that if the decay of f (r) is not too fast, then there are as. lines in Z. In the case of the Poisson Boolean model with balls of fixed radius R we characterize the critical intensity for the as. existence of lines in the covered region by an integral equation. We also determine when there are lines in the complement of a Poisson process on the space of lines in H-2.LA engOL 30